One of the great discoveries of the twentieth century is that mathematics can describe the limits of mathematical thought! We’ll discuss some of these ideas from time to time in coming weeks. In this segment, we consider Alan Turing’s insightful question:

What follows after 0, 1, 2, … , once you’ve managed to list every counting number?

Around 1875, Georg Cantor created — or discovered if you like — the transfinite ordinals : the list continues 0, 1, 2, …, then ω , ω + 1, ω + 2, etc, for quite a long long way. John H. Conway tells us about his Surreal Numbers , which add in such gems as

The stork can catch the frog even if it can start at any rational number and hop any fixed rational distance each step.

However, if the frog can start at any real number or hop any real distance, the stork has no strategy that guarantees a catch. This is, in effect, the same as proving that the real numbers are not countable.